All Questions

Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled? The boundary condition is that the ...

Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...

Let $O\subset\mathbb{R}^d$ be a bounded domain of the class $C^{1,1}$ (or $C^2$ for simplicity). Let the operator $A_D$ be formally given by the differential expression $A=-\operatorname{div}g(x)\...

Let $\kappa_1, \kappa_2>0$ be fixed. Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by $$ A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...

Let $\Omega\subset\mathbb R^n$ be a bounded domain with a sufficiently smooth boundary $\partial\Omega$. We assume $\lambda_1\in\mathbb R$ is the principle eigenvalue of the operator $$ -\Delta:\ H^...

I am studying Pazy's book "Semigroups of Linear Operators and Applications to Partial Differential Equations" and when considering an operator like: A linear differential operator, $$A : W^{...